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A Geometric Based Connection between Fractional Calculus and fractal functions

Acta Mathematica Sinica,English Series 2024年第2期40卷 537-567页

作者： Yong Shun LIANG Wei Yi SU School of Mathematics and Statistic Nanjing University of Science and TechnologyNanjing 210094P.R.China Department of Mathematics Nanjing UniversityNanjing 210023P.R.China

Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus *** the present paper,we investigate the relationship between fractional calculus and fractal functions,based only on fractal dimension *** dimension of the Riemann-Liouville fractional integral of continuous functions seems no more than fractal dimension of functions *** fractal dimension of the Riemann-Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they *** further discussion,fractal dimension of the Riemann-Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann-Liouville fractional differential is at most linearly increasing for the Holder continuous *** about other fractional calculus,such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given *** work is helpful to reveal the mechanism of fractional calculus on continuous *** the same time,it provides some theoretical basis for the rationality of the definition of fractional *** is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.

关键词： Fractional calculus fractal functions fractal dimension fractional calculus equation relationship

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Approximation of fixed points and fractal functions by means of different iterative algorithms

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CHAOS SOLITONS & fractalS 2024年 180卷

作者： Navascues, M. A. Univ Zaragoza Dept Matemat Aplicada Zaragoza Spain

Banach's contraction principle is a fundamental result in pure and applied mathematics, and all the fields of physics. In this paper a new type of contraction, called in the text partial contractivity, is presented in the framework of b -metric spaces (an extension of the usual metric spaces). The new mappings generalize the classical Banach contractions, that appear as a particular case. The properties of the new self -maps are studied, giving sufficient conditions for the existence and uniqueness of their fixed points. Afterwords three different iterative algorithms for the approximation of critical points (Picard, Ishikawa and Karakaya) are considered, concerning their convergence and stability. These findings are applied to the approximation of fractal functions, coming from contractive and non -contractive operators.

关键词： Iteration fractal functions Fixed point theorems Ishikawa algorithm b-metric spaces

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A REVIEW OF fractal functions AND APPLICATIONS

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2022年第6期30卷 2250113-2250113页

作者： Wang, Xuefei Zhao, Chunxia Yuan, Xia Nanjing Univ Sci & Technol Sch Comp Sci Nanjing 210094 Peoples R China

In this paper, we mainly investigate continuous functions with unbounded variation on closed intervals. Given the increasing number of proposals and definitions of different kinds of functions with fractal structure in the scope of fractal functions, it is important to introduce a systematic classification and discuss elementary definitions of fractal functions. Different examples of fractal functions with certain fractal dimensions have been given. Then, we have introduced different kind of fractal function such as singular fractal function, irregular fractal function, regular fractal function and complete regular fractal function. We also introduce fractal functions of integer dimension with one and two, respectively. Other continuous and discontinuous functions with fractal structure have been given. Applications of fractal functions in interpolation, approximation and relationship between fractional calculus have been investigated. Finally, we studied fractal dimension spaces with different fractal dimensions.

关键词： fractal functions fractal Dimension Variation Fractional Calculus

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Quantum Bernstein fractal functions

COMPUTATIONAL AND MATHEMATICAL METHODS

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COMPUTATIONAL AND MATHEMATICAL METHODS 2021年第3期3卷

作者： Vijender, N. Chand, A. K. B. Navascues, M. A. Sebastian, M., V Indian Inst Informat Technol Nagpur Dept Basic Sci & Engn Nagpur Maharashtra India Indian Inst Technol Madras Dept Math Chennai Tamil Nadu India Univ Zaragoza Dept Matemat Aplicada Zaragoza Spain Acad Gen Mil Ctr Univ Def Ctra Huesca S-N Zaragoza 50090 Spain

In this article, taking the quantum Bernstein functions as base functions, we have proposed the class of quantum Bernstein fractal functions. When f is an element of?(I), the base function is taken as the classical q-Bernstein polynomials, we propose the class of quantum fractal functions through a multivalued quantum fractal operator. When f is an element of Lp(I),1 <= p <= infinity, the base function is assumed as q-Kantorovich-Bernstein polynomial to construct the sequence of (q,alpha)-Kantorovich-Bernstein fractal functions that converges uniformly to Some approximation properties of these new class of quantum fractal interpolants have been studied.

关键词： q-Bernstein polynomials constrained approximation fractal functions fractals interpolation

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Associate fractal functions in L^p-spaces and in one-sided uniform approximation

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2016年第2期433卷 862-876页

作者： Viswanathan, P. Navascues, M. A. Chand, A. K. B. Indian Inst Technol Dept Math Madras 600036 Tamil Nadu India Univ Zaragoza Dept Matemat Aplicada Escuela Ingn & Arquitectura Zaragoza 50018 Spain

fractal interpolation function defined with the aid of iterated function system can be employed to show that any continuous real-valued function defined on a compact interval is a special case of a class of fractal functions (self-referential functions). Elements of the iterated function system can be selected appropriately so that the corresponding fractal function enjoys certain properties. In the first part of the paper, we associate a class of self-referential LP-functions with a prescribed LP-function. Further, we apply our construction of fractal functions in LP-spaces in some approximation problems, for instance, to derive fractal versions of the full Muntz theorems in LP-spaces. The second part of the paper is devoted to identify parameters so that the fractal functions affiliated to a given continuous function satisfy certain conditions, which in turn facilitate them to find applications in some one-sided uniform approximation problems. (C) 2015 Elsevier Inc. All rights reserved.

关键词： fractal functions L-p-spaces One-sided uniform approximation Best one-sided approximation

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fractal DIMENSION OF PRODUCT OF CONTINUOUS functions WITH BOX DIMENSION

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2023年第3期31卷 2350021-2350021页

作者： Liu, Peizhi DU, Yumeng Liang, Yongshun Nanjing Univ Sci & Technol Nanjing Sch Math & Stat Nanjing 210094 Peoples R China

This paper investigates fractal dimension of product of continuous functions with Box dimension on [0, 1]. For two continuous functions with different Box dimensions, the Box dimension of their product has been proved to be the larger one. Furthermore, the Box dimension of product of two continuous functions with the same Box dimension may not exist. Definitions of regular fractal and local fractal functions have been given. Product of a regular fractal function and a local fractal function with the same Box dimension must still be the original Box dimension.

关键词： fractal functions fractal Dimension Product of Continuous functions

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Casazza-Christensen-Eijndhoven Perturbation Result in Quasi-Banach Spaces: Application to fractal Basic Sequences

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COMPLEX ANALYSIS AND OPERATOR THEORY 2025年第3期19卷 1-20页

作者： Viswanathan, P. Indian Inst Technol Delhi Dept Math New Delhi 110016 India

A well-known classical result in functional analysis states that a bounded linear operator T on a Banach space is invertible if it is sufficiently close to the identity operator I, specifically if & Vert;I-T & Vert;<1. Later, Christensen and Casazza established a generalization of this result, which provides conditions for the invertibility of a bounded linear map on a Banach space under weaker assumptions. Recently, an effort has been made to obtain a quasi-Banach counterpart of the Christensen-Casazza perturbation result. However, this attempt missed the crucial aspect of the original result: the surjectivity of the map. The difficulty in fully extending this perturbation result stems from the fact that the dual space of a quasi-Banach space can be trivial. In this note, we circumvent this obstacle by exploring an alternative approach to the Christensen-Casazza perturbation, developed by Van Eijndhoven around the same time. Consequently, we present a complete analogue of the Christensen-Casazza-Eijndhoven perturbation result within the context of quasi-Banach. spaces. As an immediate application, we provide a fundamental sequence consisting of fractal functions for the quasi-Banach space L-p([0,1]) for 0

关键词： Quasi-Banach space Invertibility Hilding criterion Casazza-Christensen perturbation result Basic sequences fractal functions

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Taylor-Type Formulas for Arbitrary Continuous functions on Intervals and Their Application in Control Problems for Distributed Systems

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DOKLADY MATHEMATICS 2022年第2期105卷 56-60页

作者： Agadzhanov, A. N. Russian Acad Sci Trapeznikov Inst Control Sci Moscow Russia

Two classes of Taylor-type formulas for arbitrary continuous functions on intervals are obtained using Bernstein polynomials. These formulas are applicable to both smooth functions and functions that have neither finite nor infinite derivatives at any point. The Taylor-type formulas are considered in close connection with Dini derivatives, which exist for any continuous function. An example is given in which these formulas are applied to the problem of controlling a distributed oscillatory system whose dynamics obeys the d'Alembert representation.

关键词： Taylor's formula Bernstein polynomials fractal functions Dini derivatives Caputo fractional derivatives distributed systems

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fractal Convolution on the Rectangle

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COMPLEX ANALYSIS AND OPERATOR THEORY 2022年第4期16卷 1-21页

作者： Pasupathi, R. Navascues, M. A. Chand, A. K. B. Indian Inst Technol Madras Dept Math Chennai 600036 Tamil Nadu India Univ Zaragoza Dept Matemat Aplicada Escuela Ingn & Arquitectura Zaragoza 50018 Spain

The primary goal of this article is devoted to the study of fractal bases and fractal frames for L2( I x J), the collection of all square integrable functions on the rectangle I x J. The fractal function when recognized as an internal binary operation paved way for the construction of right and left partial fractal convolution operators on L2(I), for any real compact interval I. The aforementioned operators defined on one dimensional space have been applied to obtain operators on the space L2( I x J) by the identification of L2( I x J) with the tensor product space L2( I). L2(J). In this paper, we establish properties of this bounded linear operator which eventually helps to prove that L2( I x J) admits Bessel sequences, Riesz bases and frames consisting of products of fractal (self-referential) functions in a nice way.

关键词： fractal functions fractal operators fractal convolution Hilbert Spaces Schauder bases Frames Tensor Product

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Some elementary properties of Bernstein-Kantorovich operators on mixed norm Lebesgue spaces and their implications in fractal approximation

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2024年 239卷

作者： Pandey, K. K. Viswanathan, P. IIT Delhi Dept Math New Delhi 110016 India

On the one hand, the notion of mixed norm spaces has attracted considerable attention in fields such as harmonic analysis and PDE. On the other hand, a particular modification of the Bernstein operator, the so-called Bernstein- Kantorovich operator, has been of special interest for the approximation of the classical Gp-functions. This note has a double purpose. First, we record some elementary approximation properties of the Bernstein-Kantorovich operators on the mixed norm Lebesgue spaces. In the second part, we construct self-referential (fractal) counterparts to the functions belonging to the mixed norm Lebesgue spaces and introduce fractal operators on these spaces. With the help of the Bernstein-Kantorovich operators, we obtain a fractal approximation process on the mixed norm Lebesgue spaces. Furthermore, using the multivariate Haar system, we provide a Schauder basis consisting of self-referential functions for the mixed norm Lebesgue spaces, which we call the Bernstein-Kantorovich fractal Haar system. (c) 2023 Elsevier Ltd. All rights reserved.

关键词： Mixed norm Lebesgue spaces fractal functions fractal operator Bernstein-Kantorovich operators Approximation process Multivariate Haar system

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